In the array of motions possible for a single
drop or bubble moving through another fluid~
it is interesting to examine the slow steady
motion of small fluid bodies. Some experimental
data are discussed in terms of a theoretical drag
relation previously derived for low Reynolds
number flows. The need for further research
on the Tole of interfacial forces is indicated.
V. O'Brien
ike many popular ideas, the "tear drop"
shape is largely a misconception. Not that
it can never occur, but by-and-Iarge a drop of
ordinary liquid moving in another fluid has
some other shape. Moving bubbles and drops
are often globular, as in the static case, but they
may have diverse forms. The whole flow field
associated with the motion of a fluid body occurs
in such bewildering variety that a lifetime of
study devoted to such flows would probably fail
to explain them fully.
We shall concentrate here on the slow motion
of small fluid bodies, a study that will point
out both our knowledge and our ignorance.
Such fluid behavior is an immediate practical
concern of the chemical engineer in mixing and
extraction processes; it may prove to be important to aerospace engineers in understanding
the ablation of nose surfaces, in transpiration
cooling, or in problems of hydraulic systems.
A bubble is a small mass of gas in a larger
qu.antity of another fluid, either gas or liquid.
Examples are the soap bubble and an air bubble
trapped in tap water. A drop is a quantity of
liquid in another fluid, either liquid o~ gas, as
a water drop in oil or a raindrop in air. Usually
one thinks of the bubble or drop fluid as being
contained within a definite interface. However,
the inner fluid may be completely miscible with
the outer fluid and there may be no interface in
L
2
the usual sense, although there may be quite a.
sharp boundary between the fluids. The only
word the dictionary has for this situation is
blob, which includes bubble and dTop as special
cases.
There are specific reasons why a fluid blob
moving in another fluid is more interesting to
tudy than a solid moving in a fluid. A fluid
blob can deform in shape and can flow internally. There are many types of motion posible because these characteristics are controlled
by the viscous forces, the inertial pressure forces,
and the interfacial forces in rather complex ways.
Consider a very simple set of experiments.
I nto each of several large quiescent reservoirs
of the same fluid place a fluid blob having a
density different from the others. According to
the relative densities of the fluid pairs, the
earth's gravity field will cause the blobs to move
up or down. The shape of these moving blobs
will be spherical, spheroidal, hemispherical cap,
or too irregular for a simple description. The
blobs may move vertically in a straight line,
wobble back and forth in a vertical plane, spiral,
or move irregularl y. Inside the blobs there will
be either no motion, partial circulation, or a
full vortex. The over-all motion will be steady,
have an unsteady periodic component, be partially irregular, or break up entirely. Depending
on the choice of fluid pair and the size of the
APL Technical Digest
A-SPHERICAL BUBBLE. STEADY
STRAIGHT-LINE MOTION. FULL
VORTEX .
B-SPHEROIDAL FLAT-NOSED
DROP, STEADY STRAIGHT-LINE
MOTION, FULL VORTEX.
C-SPHEROIDAL FLAT-REAR DROP .
STEADY STRAIGHT-LINE MOTION, •
PARTIAL CIRCULATION.
D-SPHEROIDAL BUBBLE .
PERIODIC WOBBLE MOTION.
E-HEMISPHERICAL CAP
BUBBLE, STEADY MOTION
WITH TURBULENT WAKE.
F - DROP WITH INDENTED REAR,
STEADY STRAIGHT-LINE MOTION,
FULL CIRCULATION.
EARLY
LATER
STILL LATER
G-MISCIBLE DROP, UNSTEADY MOTION CALLED "VORTEX BREAK-UP,"
EARLY
LATER
STILL LATER
H-IMMISCIBLE DROP, UNSTEADY MOTION TO BREAK-UP,
Fig. I-A r e presentative sele ction of qua lita tive flo w p a tte rns s howing shapes and velocities of blobs, and
vortex motion.
May ·June 1962
3
blob, there can be various combinations of the
characteristics above. There can be one type
of motion at one time and another later. Figure
I illustrates qualitatively some of these cases,
with arrows showing direction of blob motion.
In general, the greater ~istortions in shape and
unsteady motion are found for the larger blob
volumes, except for the miscible cases that are
unsteady for all volumes.
The initial conditions of the experiment consist of only two fluids and a blob volume. We
might expect that such fluid fields could be explained quantitatively with a very few characteristic non-dimensional numbers; attempts to do
this, however, have only provided qualitative information. It appears that not only the usual
bulk fluid properties of viscosity and density,
but the detailed conditions at the boundary between the fluids as well, are necessary.
Bubbles
A comprehensive survey of single air-bubble
motion in various fluids was published several
years ago. 1 An attempt was made to relate the
bubble drag coefficient-an easily-obtained net
measure of the viscous retarding force-to the
three non-dimensional parameters Re, We, and
.M. Reynolds number, Re == VIp / fL' is the ratio
of inertial and viscous forces; \tVeber's number,
We == V2lp / 0-, is the ratio of inertial and surface tension forces; Rosenberg's system parameter, M == PfL4 / g0-3, is independent of bubble
size and speed. In these rela tions, V is a characteristic velocity, I is a characteristic length, g is
the gravitational acceleration constant, 0- is
the static surface tension, and p and fL are the
density and viscosity of the liquid, respectively.
Universal curves in terms of these three parameters could not be drawn from the data since a
complete description of the various flow regimes
probably depends on still-unknown interfacial
properties. It is possible, however, to relate the
measured drag of the bubble quantitatively to
certain features of the flow field. The three
important features to be discussed are internal
circulation, body shape, and Reynolds number.
·F or a spherical body and low Re, the governing Huid dynamic equation for steady motion
h as been solved. Flow patterns of the velocity
field and theoretical drag values that compare
1 W. L . Haberman and R. K. Morton, "An Experimental Investigation of the Drag and Shape of Air Bubbles Rising in
Various Liquids ," David Taylor Model Basin Rept. 802, s.~pt.
1953. (See also, Haberman and Morton, "An Experimental
Study of Bubbles Moving in Liquids," Trans. Am. Soc. Civil
Engrs., 121, 1956, 227-252.)
4
with measured drag values are found by varying
the boundary conditions imposed at the interface. The Stokes flow solution (Re<O.l) for a
spherical drop with complete circulation has
been known for half a century. Surprisingly it
does not depend on the static interfacial tension
value 0-. The tangential shearing stress is assumed to be continuous across the interface. The
shearing motion of the outside fluid would in all
likelihood cause full circulation of the viscous
fluid inside. Thus, unless the interface has some
resistance to the tangential shearing force, the
fluid body is a spherical vortex.
The Stokes drag relation is an inverse linear
function of R e for either a rigid sphere or a
fully-circulating fluid sphere. The latter case
depends on the ratio of the viscosities of the
inner and the outer fluids. Typical variation
is shown in Fig. 2 where the drag coefficient
CD for the fluid vortex sphere is less than for
the corresponding rigid sphere. Stated another
way, a circulating fluid sphere rises or falls
faster than a rigid sphere of the same density
moving in the same viscous liquid.
1000
-0
~
~IOO
1&1
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it
1&1
o
o 10
C)
«
0:
o
IO~.O~I-""""""""'O.l
I
10
100
REYNOLDS NUMBER (Red)
F'ig. 2-Theoretical drag for spherical bodies.
It is possible for a fluid blob to retain its
spherical shape beyond the low Re where the
Stokes approximations are valid. Solutions correct to the first-order in R e show that the drag
will be increased over the Stokes drag for both
the rigid and circulating spheres (Fig. 2).2
Many blobs begin to deform to an approximate oblate spheroid even at relatively low Re
(Figs. I B and C). The Stokes drag for a solid
2 V. O'Brien, "An Inves tigation of Vi scous Vortex Motion:
The Vortex Ring Cascade," (Doctoral dissertation, The Johns
Hopkins Univers ity, 1960).
APL Technical Digest
'0
~
~ IOO I::-----~-"
LLJ
U
~
u..
LLJ
o(.J
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ct
a:
o
'0.01
0.1
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10
100
REYNOLDS NUMBER (ReJ
Fig. 3-Measured drag for air bubbles.
spheroid is well known and depends on the
eccentricity of the meridian section. A flow
solution for a spheroidal vortex has been obtained recently.3 By matching tangential shear
a~ the .interfa:e, a dra?" relation for a fullyCIrculatmg fluId spherOId can be obtained. It
depends upon both the eccentricity of the crosssection and the ratio of blob viscosity to outside-liquid viscosity, IL' / IL' in 'opposing fashion.
An oblate spheroid always has higher drag than
a sphere of the same volume, drag increasing
as the eccentricity increases. As with the fluidvortex sphere, the circulating spheroid has less
drag than the rigid body. The drag depends
on the viscosity ratio, being least for vanishing
IL' / IL· The effect is a translation upward of the
three curves in Fig. 2 for a given spheroidal
body.
. D~ta for air bubbles rising in four typical
lIquIds (from Ref. 1) are shown in Fig. 3. Without resistive, tangential, interfacial forces acting
on these bubbles, one would expect full circulation. Because of the low gas viscosity, the data
should lie on the lowest curve of Fig. 2 during
the period when the shape is spherical. As Re
increases, the ~hape is likely to become more
spheroidal, with a resultant upward shift in
drag (deformation depending on the resistance
to the deforming normal pressure forces). At
the very lowest Re, which is the Stokes flow
regime, air bubbles seem to be rigid; the drag
is just Stokes drag for a rigid sphere. Although
Ref. 1 did not reveal circulation patterns within
the bubbles directly, other work has shown that
• 3 V. O ' Brien, "Steady Spheroidal Vortices- More Exact Solur~o6~~ ~~3:f:8 .Navier-Stokes Equation," Quart . Appl. Math., 19,
M ay -June 1962
they do exist, but only for the higher Re. At a
certain Re, the circulation appears first at the
front of the fluid parcel (Fig. 1C). As Re increases, the circulation covers a greater portion
of the interior until the whole is circulating. It
s:ems e~ident that the bubble in the syrup is
CIrculatIng at very low Re; the bubble in oil
begins to circulate later and shows deformation
effects at an Re of 1.0. Circulation within the
bubble in either water or the corn syrup solution is inhibited throughout this low Re range.
There is no parameter yet to explain this
difference in circulation. Without doubt, both
circulation and deformation are intimately connected with the dynamic characteristics of the
interface. Unfortunately, however, studies of
these properties are spotty, and suitable data
are very sparse.
Drops
Data similar to those for bubbles have been
presented for drops moving in liquids for a
number of liquid pairs. 4 ,5 Unhappily, the overall pattern of behavior seems to have more variety than the gas bubbles at the higher Reynolds
numbers. The differences are probably d..ue to
the range of interfacial tensions of these systems
while the gas-liquid pair almost invariably has a
high interfacial tension value. However, in the
lower Re ranges the behavior is remarkably
similar. Very tiny drops do not circulate at all,
and measured drags lie on the Stokes rigidsphere line. As Re increases, circulation begins
inside the drops and the drag values drop from
the rigid to the fully-circulating fluid-sphere
curve. If deformation takes place, the drag
curves rise above the sphere values .
For the fluid vortex sphere, the dip of the
drag curve below the rigid-sphere curve depends
on the viscosity ratio of the system. For comparison with air bubble data in Fig. 3, consider only the drop systems where the viscosity
ratio IL' / IL ~ O. Figure 4 shows some data on
tetrachloroethylene, bromobenzene, and ethylbromide drops through 80-100% glycerin, and
tetrachloroethylene through a 68 % corn syrup
solution. These drops seem to be fully circulating at small Re. The validity of the first-order
approximation theory is restricted to Re L. 1,
but apparently it is an overestimate for
1 < R e L. 10. Thus, the first-order drag rela4 M. Warshay et al, " Ultimate Velocity of Drops in Stationary Liquid Media," Can. J . Chem. Eng., 37, 1959, 29-36 .
5 R . Satapathy and W. Smith, "The Motion of Single lm~t~~~~ Drops Through a Liquid," J. Fluid Meeh., 10, 1961,
5
~OO r---~~~-r'-rrrrr---~--'--r'-~~r---~--'-'-'-~~
l rr
l ----,-~I--~-r~~
TETRACHLOROETHYLENE IN CORN SYRUP (Ref 4)
TETRACHLOROETHYLENE IN GLYCERIN (Ref 4)
TETRACHLOROETHYLENE IN GLYCERIN (Ref 5)
BROMO BENZENE IN GLYCERIN (Ref 5)
ETHYLBROMIDE IN GLYCERIN (Ref 5)
A
o
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<>
_ IOO ~--------------~~r-~--------+---------------4---------------~
o
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t-
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..............
•...••...·RIGID SPHERE
I
I
I
"
I
100
REYNOLDS NUMBER (R~)
Fig. 4-Measured drag for falling spherical drops.
tion and the Stokes drag relation 6 provide upper
and lower bounds on the actual drag in this
range.
Blobs
In the absence of an interface, the blob motion
is always unsteady. The importance of the
vortex pattern, and the change with time in
appearance of a liquid blob until it ultimately
breaks up (Fig. IG), have been described before. 7 By increasing the drop volume sufficiently,
in the case of immiscible falling drops, a point
is often reached where the drop breaks up. The
deformation and breakup for: these drops are
quite similar to the miscible blob behavior (see
Fig. IH and Ref. 7). It is very likely that the
"vortex-breakup" behavior depends on the
tangential, dynamic, interfacial characteristics.
6 M. J. Hadamard, "Mouvement permanent lent d'une sphere
liquide et visqueuse dans un liquid visqueux," Compt. rend.,
152, 1911, 1735-1738.
7 V. O'Brien, "Why Raindrops Break Up-Vortex Instability,"
J. M et eorol., 18, 1961, 549-552.
6
While no attempt has been made to cover
fully all the aspects of the moving blob problem,
the importance of the influence of the interfacial
characteristics on the qualitative regimes of blob
motion has been stressed. The main points of
the low Re flow about bubbles, drops, and other
blobs can be summarized:
1. Viscosity and shear cause circulation.
2. Circulation causes drag decrease (velocity
rise).
3. Inertial effects cause drag rise.
4. Pressure forces cause change in body shape
and, thus, further drag rise.
5. Interfacial forces oppose transmission of
tangential shear and, therefore, oppose
circula tion.
6. Interfacial forces resist change in shape.
Much more research on the role of interfacial
forces must be done before a quantitative explanation of the onset of circulation and the
deformation of the blob can be obtained.
APL Technical Digest